The support poset of a monomial ideal I⊆k[x1,…,xn] encodes the relation between the variables x1,…,xn
Nearby in the stack
and the minimal monomial generators of
I
. It is known that not every poset is realizable as the support poset of some monomial ideal. We describe some posets
P
for which we can explicitly find at least one monomial ideal
IP
such that
P
is the support poset of
IP
. Also, for some families of monomial ideals we describe their support posets and study their properties. As an example of application we examine the relation between forests and series-parallel ideals.